Uncertainty Quantification for Forward and Inverse Problems of PDEs via Latent Global Evolution

Deep learning-based surrogate models have demonstrated remarkable advantages over classical solvers in terms of speed, often achieving speedups of 10 to 1000 times over traditional partial differential equation (PDE) solvers. However, a significant challenge hindering their widespread adoption in both scientific and industrial domains is the lack of understanding about their prediction uncertainties, particularly in scenarios that involve critical decision making. To address this limitation, we propose a method that integrates efficient and precise uncertainty quantification into a deep learning-based surrogate model. Our method, termed Latent Evolution of PDEs with Uncertainty Quantification (LE-PDE-UQ), endows deep learning-based surrogate models with robust and efficient uncertainty quantification capabilities for both forward and inverse problems. LE-PDE-UQ leverages latent vectors within a latent space to evolve both the system's state and its corresponding uncertainty estimation. The latent vectors are decoded to provide predictions for the system's state as well as estimates of its uncertainty. In extensive experiments, we demonstrate the accurate uncertainty quantification performance of our approach, surpassing that of strong baselines including deep ensembles, Bayesian neural network layers, and dropout. Our method excels at propagating uncertainty over extended auto-regressive rollouts, making it suitable for scenarios involving long-term predictions. Our code is available at: https://github.com/AI4Science-WestlakeU/le-pde-uq.

基于深度学习的代理模型在速度方面相较于经典求解器展现出显著优势，通常比传统的偏微分方程（PDE）求解器快10到1000倍。然而，阻碍它们在科学和工业领域广泛应用的一个重大挑战是对其预测不确定性缺乏了解，特别是在涉及关键决策的场景中。为解决这一局限，我们提出一种将高效且精确的不确定性量化集成到基于深度学习的代理模型中的方法。我们的方法被称为具有不确定性量化的偏微分方程潜在演化（LE - PDE - UQ），它为基于深度学习的代理模型赋予了针对正问题和逆问题的强大且高效的不确定性量化能力。LE - PDE - UQ利用潜在空间内的潜在向量来演化系统状态及其相应的不确定性估计。潜在向量被解码以提供系统状态的预测以及其不确定性的估计。在大量实验中，我们证明了我们的方法具有准确的不确定性量化性能，超过了包括深度集成、贝叶斯神经网络层和随机失活在内的强大基准方法。我们的方法在扩展的自回归展开中能出色地传播不确定性，使其适用于涉及长期预测的场景。我们的代码可在以下网址获取：https://github.com/AI4Science - WestlakeU/le - pde - uq